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One Sample t Test

1.1. The t distribution

One Sample t test

An extension of the Z-test with a sample mean.
- Still comparing a sample mean to a population mean.
But, we have the problem of an unknown $\sigma$ and $\sigma^2$ .
Both $S$ and $S^2$ are biased because samples are by their very nature, smaller than the population(s) they represent.
So, we apply a progressive correction ( $n - 1$ ), to arrive at an unbiased estimate of the population variance.
$S^2 = \frac{\sum\left(X - \overline{X}\right)^2}{n - 1}$
We say progressive correction because; as sample size () gets larger, the correction has less effect.
- Larger sample $\rightarrow$ includes more of the population $\rightarrow$ less bias.
- Therefore, less correction is needed as sample size increases.

Degrees of Freedom (df)

We call this progressive correction the degrees of Freedom (df).

The number of scores in a sample which are ``free to vary''.
- If you know the mean and all but one of the scores, you can figure the score you don't know.
- So, for one sample t test: $df = n -1$
Now we can go on with some of the `usual stuff'.
- Distribution of Means...
$S_M^2 = \frac{S^2}{n}$ $S_M = \sqrt{S_M^2}$

Comparison Distribution Changes

Because we are estimating the population variance, we use the t distribution (instead of the Z distribution as was done previously).
- The t distribution is not normally distributed, due to more error in estimation (i.e., estimating the unknown $\sigma^2$ ).
Different table, similar procedure for finding the cutoff sample score associated with , or any other significance level.
- But, you must take into account sample size; meaning, the df in order to find $t_{crit}$

An Excerpt from the t distribution table

Table 1: t Critical Values

1-tailed	df	.10	.05	.025	.01
2-tailed	df	.20	.10	.05	.02
	1	3.078	6.314	12.710	31.821
	2	1.886	2.920	4.303	6.965
	3	1.638	2.353	3.182	4.541
	4	1.533	2.132	2.776	3.747
	5	1.476	2.015	2.571	3.365
	6	1.440	1.943	2.447	3.143
	etc.

Degrees of freedom in the left column and significance level along the top rows.

Calculating the One Sample t

Also referred to as $t_{calc}$ which is compared to $t_{crit}$
Recall, $S_M$ is the standard deviation of a distribution of means.
First, calculate $S_M$ ( $\mu$ is given): $S^2 = \frac{\sum{\left(X - \overline{X}\right)^2}}{n - 1}$ then
$S_M^2 = \frac{S^2}{n}$ then
$S_M = \sqrt{S_M^2}$
Which leads to:
$t_{calc} = \frac{\overline{X} - \mu}{S_M} = \frac{\overline{X} - \mu}{S/\sqrt{n}} = \frac{\overline{X} - \mu}{\sqrt{S^2/n}}$
- The formula below is the one used here but, all three above are equivalent ( $t_{calc}$ ).
  $t_{calc} = \frac{\overline{X} - \mu}{S_M}$

1.2. One Sample t-test Example

NHST Example

Research question:

Does this class of four students get less sleep than all UNT students ( $\mu = 6.08$ hours)?
Step 1: Define the populations and restate the research question as null and alternative hypotheses.
Population 1: Students in this class.
Population 2: All other UNT students.
$H_0: \mu_1 \geq \mu_2$
$H_1: \mu_1 < \mu_2$
Note, the Alternative hypothesis ( $H_1$ ) is directional: one-tailed test.

Step 2: Comparison Distribution

Table 2: Comparison dist. (estimate $\sigma$ with $S_M$ )

$X$	$\overline{X}$	$X - \overline{X}$	$\left(X - \overline{X}\right)^2$
5	4	1	1
3	4	-1	1
6	4	2	4
2	4	-2	4
16			$SOS = 10$

$S^2 = \frac{\sum{\left(X - \overline{X}\right)}^2}{n - 1} = \frac{10}{4 - 1} = 3.33$

$S_M^2 = \frac{S^2}{n} = \frac{3.33}{4} = 0.8325$

$S_M = \sqrt{S_M^2} = \sqrt{0.8325} = 0.912$

Step 3: Determine the critical score

Determine the cutoff sample score.
Significance level = .05, one-tailed test, $df = 4 - 1 = 3$
Look in the appropriate column of the t table.
- One-tailed, significance level = .05
http://www.math.unb.ca/~knight/utility/t-table.htm
Critical t value is 2.353
Which means, we need $t_{calc}$ to be greater than |2.353| (absolute value of 2.353) to find a significant difference.

Step 4: Calculate t ( $t_{calc}$ )

From previous calculating, we have:
$\overline{X} = 4.00$ , $S_M = 0.912$
So, we can plug in the values, including the population mean which was given.
$t_{calc} = \frac{\overline{X} - \mu}{S_M} = \frac{4.00 - 6.08}{0.912} = \frac{-2.08}{0.912} = -2.28$

Step 5: Compare and Make a Decision

Since $t_{calc} = \vert-2.28\vert < \vert 2.353\vert = t_{crit}$ we fail to reject the null hypothesis.
The interpretation is...Students in this class ( $M = 4.00$ ) do not sleep significantly fewer hours than all other students at UNT ( $\mu = 6.08$ ), $t(3) = -2.28, p > .05$ .
Do not be tempted to say something like; nearly significant, just missed significance, etc.
- Although the sample mean is numerically smaller, it is not statistically significantly smaller.
- Remember, we have a very small sample size, so we would need a very large mean difference to achieve significance (or a larger sample).

1.3. Effect Size

Effect Size

The appropriate effect size measure for the one sample t test is Cohen's d.
Calculation of d in its general form (as it was with the Z-test) is:
$d = \frac{\overline{X} - \mu}{\sigma}$
However, we do not know the population standard deviation ( $\sigma$ ) in the t situation, so we estimate with $S = \sqrt{S^2}$
$d = \frac{\overline{X} - \mu}{S} = \frac{4 - 6.08}{\sqrt{3.33}} = \frac{-2.08}{1.82} = 1.143$
So, although we have a large effect size (standardized difference), we did not achieve statistical significance. However, keep in mind that with a larger sample, this amount of mean difference may have been significant.
- Statistical significance is directly tied to sample size, effect size is not.

1.4. Using Delta for Statistical Power

Statistical Power and Sample Size

Calculating power with the one sample t test is slightly different than with the Z-test.
Here, we introduce something called the Non-Centrality Parameter (NCP).
- The NCP takes its name from the fact that if is false, the distribution of our statistic will not be centered on zero, but on something called delta.
  - Thus, the distribution is non-central.
  - The symbol for delta is: $\delta$
In the context of the one sample t test, delta is calculated as follows:
$\delta = d * \sqrt{n}$

Statistical Power and Sample Size

To calculate delta ( $\delta$ ) for our current example:
$\delta = 1.143 * \sqrt{4} = 1.143 * 2 = 2.286$
With the significance level (.05, one-tailed), we can look up power in a $\delta$ table.
which is not terribly meaningful once the study has been completed.

However, we can now use to calculate sample size a-priori for a given power (prior to collecting data for the next study). Where This type of power calculation (figuring a-priori sample size) is very useful.

1.5. Calculating a Confidence Interval Confidence intervals allow us to go beyond NHST and offer us an idea of the location of a population mean (). We need three numbers to calculate a confidence interval: The Standard Error (SE) which is also known as the standard deviation of a distribution of means: The critical value from our current study: And the mean of our sample: Recall the general formulas for a confidence interval from Module 6: Calculating a CI In this example, we are calculating a 95% confidence interval () because our critical value was based on a significance level of .05 (1 - .05 = .95 or 95%). , , If we were to take an infinite number of samples of students in this class, 95% of those samples' means would be between 6.146 and 1.854 hours of sleep. Remember, the population mean is fixed (but unknown); while each sample has its own mean (sample means fluctuate). Considerations and Interpretations The current example resulted in . Notice the population 2 mean (), representing all UNT students, falls inside our interval. Recall, we did not reject the null; meaning our sample did come from population 2, not a distinct population (i.e. population 1). If were greater than 6.146, then we would have rejected the null hypothesis and inferred that our sample came from population 1; meaning, population 1 would have been significantly different from population 2. Important: the interval is not interpreted as ``we are 95% confident that population 1's mean is between 6.146 and 1.854.'' We are dealing with a sample; we do not know what is; so, we can not know what that interval would be. 1.6. Summary of Section 1 One Sample t test Usage? Unfortunately... The one sample t test is essentially never used, but it servers a good purpose to familiarize us with the t distribution. Occasionally, we know both and , for instance with SAT or GRE scores, which necessitates the Z-test. Generally, we do not know or so the one sample t test is not frequently used. Instead, as we will see; we estimate population values with sample statistics and compare samples to infer effects in the general population(s) of interest. The t distribution is used for other types of t tests which will be covered shortly. Summary of Section 1: One Sample t Test Section 1 covered the following topics: Introduced the t distribution One Sample t test Cohen's d Effect Size Use of delta () for Statistical Power Use of delta () for calculating a-priori sample size Calculation of Confidence Intervals. Next: Dependent Up: Module 8: Introduction to Previous: Contents Contents jds0282 2010-10-15